The Fundamental Theorem of Calculus

*Let’s investigate the Fundamental Theorem of Calculus using programs written for
the *CASIO* CFX-9850G.*

Let’s look at two forms of this theorem.

__First Form__: Let f(x) be an integrable function defined on [a, b] and let F(x) also
be defined on [a, b] such that F(x) is continuous on [a, b] and the derivative F(x) is
f(x) for all x in (a,b). Then

ò _{a}^{b} f(x) dx = F(b) -
F(a)

__Second Form__: Let f(x) be an integrable function defined on [a, b] and let F(x) also
be defined on [a, b] by

F(x) = ò _{a}^{b} f(x) dx +
F(a) for all x in [a, b]

where the real number F(a) is specified in advance. Then, F(x) is continuous on [a, b], and if f(x) is continuous at a point c in [a, b] then F(x) is differentiable at c also and F(c) = f(c)

We will use this second form and the **CASIO CFX-9850G**. to

(1) Choose an integrable (and continuous) function f(x) and convenient viewing window.

(2) Construct the __numerical__ __antiderivative__ F(x) = f(x) dx over an
appropriate domain.

(3) Construct the __numerical__ __derivative__ of F(x) in (2) above.

(4) Graphically compare the numerical derivative of the numerical antiderivative of F(x) with the original function f(x) to see equality.

Obtain the following programs from the instructor by linking your calculators and transferring programs or key the programs below into the calculators memory:

Program FTC

ViewWindow -5, 5, 1, -5, 5, 1

Cls

Graph Y = Y1

StoPict 3

Cls

Seq(x, x, 1, 127, 1) --> List 1

List 1 -- List 2

"F(A) = " ? --> F

"A = " ? --> A

(Xmax - Xmin) / 127 --> D

1 --> B

Lbl 1

(Y1, A, Xmin+BD) --> C

Xmin+BD --> List1[B]

C --> List2 [B]

PlotOn Xmin+BD, C+F

Isz B

B __<__ 126

Goto 1

StoPict 1

Program NDERIVE

Cls

List 1 --> List 3

For 2 --> I To 126

(List 2[I] - List 2[I-1]) / (List 1[I] - List 1[I-1]) --> List 3[I]

PlotOn List 1[I], List 3[I]

Next

StoPict 2

Stop

After receiving these two programs from us, follow these steps:

(1) Return to the MAIN menu screen.

(2) Enter GRAPH mode and input the function Y1 = X^{2}.

(3) Press **MENU ALPHA log** (B). You will see FTC and NDERIVE in the program list.
Highlight FTC and press **F1** (EXE) to execute this program.

(4) The program graphs Y1 in the viewing window defined by the program. After viewing,
press **EXE** to continue program execution.

(5) You will be prompted for two input values.

At the prompt "F(A)=?", enter zero and press **EXE**.

At the prompt "A=?", enter zero and press **EXE** again.

(6) The screen is momentarily blank, but soon the numerical antiderivative of Y1 is plotted. (This plot is also saved in a picture memory that we will see later for comparison.)

(7) Press **SHIFT** **EXIT** (QUIT) **MENU**. Press **4** to see the
x-values in List 1 and corresponding y-values in List 2 for the numerical derivative just
plotted.

(8) After viewing these lists, press **MENU ALPHA log** (B). Highlight program
NDERIVE and press **F1** (EXE) to execute it. Soon, the numerical derivative is plotted
from the data saved by program FTC into List 1 and List 2. This is the numerical
derivative of the numerical antiderivative of Y1.

(9) Now press **OPTN** **F1** (PICT) **F2** (RCL) **F1** (Pict1) to
superimpose the numerical antiderivative of Y1.

(10) Press **OPTN** **F1** (PICT) **F2** (RCL) **F3** (Pict3) to also
superimpose the graph of Y1 on the screen. Look carefully.

Your Turn

Try these programs on the following functions. Be sure to first place into Y1 in GRAPH mode. In each case set, "F(A) = 0" and "A = 0".

(1) Y1 = x

(2) Y1 = x(x+1)(x-2)

__Note__: Trigonometric functions will not work well with program FTC as it is written.
The View Window settings in the program must be appropriately set for these types of
functions.

Dual Screen

A useful feature of the **CASIO CFX-9850G** is its ability to display split viewing screens so that
different graphs can be viewed at the same time.

Use dual screen to view f(x) = x ( x + 2 )( x - 1 ) and g(x) = 1 - 2 x^{2}

Turn on the **CASIO CFX-9850G** by pressing

Use the cursor arrows to highlight the GRAPH menu and press **EXE **or just press **5**
when at the main menu screen.

Press **SHIFT** **MENU** (SETUP) and use the down cursor arrow to highlight the
Dual Screen line. Press F1 (Grph). Press the **EXIT** key to return to the function
list.

Press **SHIFT** **F3** (V-Window) to set the parameters for the __left screen__.
Use the following:

Xmin = -3 | Xmax = 3 | Xscale = 0.5 |

Ymin = -3 | Ymax = 3 | Yscale = 1 |

Press **F6** (RIGHT) to set the parameters for the __right screen__. Use the
following:

Xmin = -5 | Xmax = 5 | Xscale = 1 |

Ymin = -4 | Ymax = 4 | Yscale = 1 |

Press **EXIT** to return to the function list and enter into the Y1 slot** **X (
X + 2 )( X - 1 ) and into the Y2 slot 1 - 2 X^{2}

Deselect Y1 by pressing **F1** (SEL) and graph Y2 on the left screen by pressing **F6**
(DRAW).

Press **OPTN** **F2** (SWAP). This interchanges the left screen which is the only
active screen with the right screen which is inactive. Press **EXIT** to return to the
function list and note the __reverse video R__ appearing on the Y2 function line
indicating that this is on the right screen.

Now select Y1 by highlighting if and pressing **F1** (SEL). Press **F6** (DRAW)
to view the graph on the active left screen.

We can view each screen separately by pressing **F6** (G<->T) for the active
screen (the left screen) and **F6** (G<->T) again for the inactive screen (the
right screen).

Press **EXIT** or **F6** (G<->T) to return to the function list. Press **F6**
(DRAW) to view both screens again and now press **OPTN** **F1** (COPY). This
superimposes Y2 on Y1 on the inactive screen.

Press **EXIT** to return to the function list and note that Y1 has a __reverse
video B__ on its function line indicating that it is plotted on both view windows.

Use the cursor arrows to highlight Y2 and press **F2 **(DEL)** F1** (YES).

Use the cursor arrows to highlight Y1 and press **F1** (SEL) to deselect Y1 removing
it from both screens and press **F1** (SEL) to select Y1 for graphing onto the active
left screen.

Press F6 (DRAW) to see Y1 on the left screen.

The split screen feature is especially useful when using the other graphing features.

Press **SHIFT** **F2** (ZOOM) **F1** (BOX) and expand a box around the local
maximum located at x = -2 -- this is accomplished by using the cursor arrows to move the
"cross hairs" to a position marking the top left-hand corner of the desired box
and pressing **EXE** then moving down to mark the bottom right-hand corner of the box
by pressing **EXE** again. The desired box is magnified and place into the inactive
right screen. Practice these steps for the local minimum at x = 1.

Press **EXIT** to return to the function list and **F2** (DEL) **F1** (YES) to
delete Y1.

Press **MENU**.

Turn off the **CASIO CFX-9850G** by pressing

Graph to Table

Do a polar coordinate plot of r = 2 sin( q ) - cos( 2 q ) and store the values of the graph at 0° , 36° , 72° , 90° , 108° , 144° , 180° , 216° , 270° , 324° , and 360° into a table.

The Graph-to-Table feature of the **CASIO CFX-9850G** allows us to show both a graph and a table associated with
the graph. We can move the "cross hairs" on a graph and store its values into a
corresponding table.

Turn on the **CASIO CFX-9850G** by pressing

Use the cursor arrows to highlight the **GRAPH** menu and press **EXE** or just
press **5** when at the main menu screen.

Press **SHIFT** **MENU** (SETUP) and use the down cursor arrow to highlight the
Dual Screen line. Press **F2** (GtoT). Continue to scroll down to the Angle line and
press **F1** (Deg). Press **EXIT** to return to the function list.

Press **SHIFT** **F3** (V-Window) to set the parameters for the __left screen__.
Use the following:

Xmin = -2 | Xmax = 2 | Xscale = 0.5 |

Ymin = -1 | Ymax = 4 | Yscale = 0.25 |

Tmin = 0 | Tmax = 360 | Tpitch = 3.6 |

Press **EXIT** to return to the function list.

Press **F3** (TYPE) **F2** (r=) and enter 2 sin( q ) -
cos( 2 q ) into slot r1. Press **EXE** to store the
function.

Press **F6** (DRAW). The graph appears on the left screen and an empty table appears
on the right screen.

Press **SHIFT F1** (Trace) and use the right cursor arrow to move the cross hairs to
0° . Press **EXE** to place the r and q
values into the table on the right screen. Continue moving the cross hairs and storing the
desired values.

The table appears as:

r1 | q |

0 | -1 |

36 | 0.8665 |

72 | 2.7111 |

90 | 3 |

108 | 2.7111 |

144 | 0.8665 |

180 | -1 |

216 | -1.484 |

270 | -1 |

324 | -1.484 |

360 | -1 |

Do an xy-plot of y = x^{3} and y = x + ^{1 }/ _{x} and create a
table of their point of intersection and the local minimum of the second function.

Press **EXIT** to return to the function list.

Press **F3** (TYPE) **F1** (Y=) and enter X^3 into slot Y2. Press **EXE** to
store the function.

Enter X + 1 ¸ X into slot Y3 and press **EXE** to store
the function.

Press **SHIFT** **F3** (V-Window) to set the parameters for the left screen. Use
the following:

Xmin = 0 | Xmax = 4 | Xscale = 0.5 |

Ymin = 0 | Ymax = 4 | Yscale = 0.5 |

Press **EXIT** to return to the function list.

Press **F6** (DRAW). The graph appears on the left screen and an empty table appears
on the right screen.

Use the Zoom Box feature to get a better view of the intersection. Press **SHIFT** **F2**
(ZOOM) **F1** (BOX), move the cross hairs to the top left-hand corner of a desired box,
and press **EXE**. Now move the cross hairs to the bottom right-hand corner of the
desired box and press **EXE**. Press **SHIFT F1** (Trace) and use the right cursor
arrow to move the cross hairs to the point of intersection of Y2 and Y3. Use the up and
down cursor arrows to view the point of intersection from the prospective of each
function, for example from Y2 the intersection is (1.271592..., 2.056096...) and from Y3
the intersection is (1.271592..., 2.058007...)

Press **EXE** to place the X and Y values into the table on the right screen.

Press **SHIFT** **F2** (ZOOM) **F6** ( |> ) **F1** (ORIG) to return to
the original graph.

Press **SHIFT** **F1** (TRACE) and use the up or down arrow to focus on Y3. Now
use the right cursor arrow to move to the local minimum. Press **EXE** to store that
point into the table on the right screen. The local minimum values are approximately
(1.032258..., 2.001008...).

Press **SHIFT** **F1** (TRACE) **AC/ ^{ON}** to return to the table.

To save the X tabled data into List 1 press **OPTN** **F2** (LMEM) **F1**
(List1).

Use the right cursor arrow to highlight the Y2 tabled data. To save it into List 2,
press **OPTN** **F2** (LMEM) **F2** (List2).

Use the right cursor arrow to highlight the Y3 tabled data. To save it into List 3,
press **OPTN** **F2** (LMEM) **F3** (List3).

Press **EXIT** **MENU** **4** and see that the table data has been transferred
into these lists.

Press **MENU**.

Turn off the **CASIO CFX-9850G** by pressing

Dynamic Graphing

Animate the changes produced by the angle of the parametric graph given by

X_{t} = ( 10 cos a ) t and Y_{t}
= ( 10 sin a ) t - 4.9 t^{2}

The Dynamic Graphing feature of the **CASIO CFX-9850G** allows us to display real-time representations of changes in
a graph as coefficients are changed.

Turn on the **CASIO CFX-9850G** by pressing

Use the cursor arrows to highlight the **DYNA** menu and press **EXE** or just
press **6** when at the main menu screen.

Press **SHIFT** **MENU** (SETUP) and use the down cursor arrow to highlight the
Angle line. Press **F1** (Deg). Press **EXIT** to return to the function list.

Deselect any functions in the function list by using the up or down cursor arrow to
highlight the selected function and pressing **F1** (SEL). We can also delete any
functions that appear by again highlighting them and pressing **F2** (DEL) **F1**
(YES).

Press **SHIFT** **F3** (V-Window) to set the parameters. Use the following:

Xmin = 0 | Xmax = 15 | Xscale = 2 |

Ymin = -1 | Ymax = 4 | Yscale = 0.25 |

Use the down cursor arrow to scroll down to enter the following:

Tmin = 0 Tmax = 2 Tpitch = 0.2

Press **EXE** to return to the function list. Press **F3** (TYPE) **F3**
(Parm). Notice that the function list changes to accommodate parametric functions (Xt,
Yt).

Enter into slot Xt1 the expression** **( 10 cos A ) T. Note that the variable A is
obtained by pressing **ALPHA X,q ,T** (A). Press **EXE**
to store the function.

Enter into slot Yt1 the expression ( 10 sin A )T - 4.9 T^{2}. Press **EXE**
to store the function.

Press **F4** (VAR), highlight A, enter the value 30, and press **EXE**. Press **F2**
(RANG) and set the parameters as follows:

Start: 30 | End: 75 | pitch: 15 |

Press **EXIT** to return to the previous screen.

Press **F3** (SPEED) and use the up or down cursor arrow to highlight the Normal
line then press **F1** (SEL). Press **EXIT** to return to the previous menu.

Press **F6** (DYNA) to see the animation. This animation will continue through ten
iterations or press **AC/ ^{ON}** to stop the process at any point and return to
the Dynamic Range screen. The speed of the presentation can be controlled from this
screen. The menu selections are:

**F1** ( |||> ) - Stop and go speed. With each **EXE** keystroke one graph is
displayed.

**F2** ( > ) - Slow speed presentation.

**F3** ( |> ) - Normal speed presentation.

**F4** ( >> ) - Fast speed presentation.

**F5 **(STO) - Stores graph settings and screen data.

**F6 **(DEL) - Delete Dynamic Graph screen data.

Press **EXIT** to return to the Dynamic Var screen and press **EXIT** again to
return to the function list.

Animate the changes produced by the leading coefficient of the function

f(x) = b (x - 7)^{2} + 1.

Deselect any functions in the function list by using the up or down cursor arrow to
highlight the selected function and pressing **F1** (SEL). We can also delete any
functions that appear by again highlighting them and pressing **F2** (DEL) **F1**
(YES).

Keep the same viewing window settings as above.

Press **F3** (TYPE) **F1** (Y =) and enter into the Y2 slot the expression B ( X
- 7 )^{2} + 1. Note that the variable B is obtained by pressing **ALPHA log**
(B). Press **EXE** to store the function.

Press **F4** (VAR), highlight B, enter the value 0.25, and press **EXE**. Press **F2**
(RANG) and set the parameters as follows:

Start: 0.25 End: 4.25 pitch: 1

Press** EXIT** to return to the previous menu.

Press **F6** (DYNA) to see the animation. Press **AC/ ^{ON}** to halt the
presentation and press

Now press **SHIFT MENU** (SETUP) and use the down cursor arrow to highlight the
Locus line. Press **F1** (On). Press **EXIT **to return to the Dynamic Var screen.

Press **F6** (DYNA) to get a locus of graphs that change colors. Press **AC/ ^{ON}**
to stop the presentation and return to the Dynamic Range screen. Press

Use the up or down cursor arrow to highlight each function and press **F2** (DEL) **F1**
(YES) to remove each from the function list.

Press **MENU**.

Turn off the **CASIO CFX-9850G** by pressing